Convergence rates in homogenization of higher order parabolic systems

This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp O(\va) convergence rate in the space L^2(0,T; H^{m-1}(\Om)) is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by \cite{suslinaD2013} is used here.Comment: 28 page

Blow-up phenomena for a family of Burgers-like equations

AbstractBy introducing a stress multiplier we derive a family of Burgers-like equations. We investigate the blow-up phenomena of the equations both on the real line R and on the circle S to get a comparison with the Degasperis–Procesi equation. On the line R, we first establish the local well-posedness and the blow-up scenario. Then we use conservation laws of the equations to get the estimate for the L∞-norm of the strong solutions, by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles. Analogous results are provided in the periodic case. Especially, we find differences between the Burgers-like equations and the Degasperis–Procesi equation, see Remark 4.1

Well-posedness and regularity of the Darcy-Boussinesq system in layered porous media

We investigate the Darcy-Boussinesq model for convection in layered porous media. In particular, we establish the well-posedness of the model in two and three spatial dimension, and derive the regularity of the solutions in a novel piecewise H2 space

Low-complexity full-field ultrafast nonlinear dynamics prediction by a convolutional feature separation modeling method

The modeling and prediction of the ultrafast nonlinear dynamics in the optical fiber are essential for the studies of laser design, experimental optimization, and other fundamental applications. The traditional propagation modeling method based on the nonlinear Schr\"odinger equation (NLSE) has long been regarded as extremely time-consuming, especially for designing and optimizing experiments. The recurrent neural network (RNN) has been implemented as an accurate intensity prediction tool with reduced complexity and good generalization capability. However, the complexity of long grid input points and the flexibility of neural network structure should be further optimized for broader applications. Here, we propose a convolutional feature separation modeling method to predict full-field ultrafast nonlinear dynamics with low complexity and high flexibility, where the linear effects are firstly modeled by NLSE-derived methods, then a convolutional deep learning method is implemented for nonlinearity modeling. With this method, the temporal relevance of nonlinear effects is substantially shortened, and the parameters and scale of neural networks can be greatly reduced. The running time achieves a 94% reduction versus NLSE and an 87% reduction versus RNN without accuracy deterioration. In addition, the input pulse conditions, including grid point numbers, durations, peak powers, and propagation distance, can be flexibly changed during the predicting process. The results represent a remarkable improvement in the ultrafast nonlinear dynamics prediction and this work also provides novel perspectives of the feature separation modeling method for quickly and flexibly studying the nonlinear characteristics in other fields.Comment: 15 pages,9 figure